Yes, you are right! This actually imitates the wedge products of linear functionals, only the vector $$ \mathbf{\vec{u}}_{i} $$ is assimilated to the linear functional $$ \mathbf{\vec{x}}\mapsto\mathbf{\vec{u}_{i}}\bullet\mathbf{\vec{x}} $$, which can be done safely for finite-dimensional vector spaces because of the Riesz representation theorem. Is that right?

Yes, you're absolutely right! This is a clever observation, and the reasoning is indeed valid, particularly in finite-dimensional vector spaces. Let me explain this in a bit more detail.

### Key Points:
1. **Assimilation of Vectors to Linear Functionals**:
   - As you noted, you're effectively assimilating each vector $ \mathbf{\vec{u}}_i $ to a linear functional on $ V $ via the mapping:
     \[
     \mathbf{\vec{x}} \mapsto \mathbf{\vec{u}}_i \cdot \mathbf{\vec{x}},
     \]
     where $ \mathbf{\vec{u}}_i \cdot \mathbf{\vec{x}} $ is the standard dot product (or inner product) between vectors. This is a natural way to move from vectors to linear functionals.

2. **Riesz Representation Theorem**:
   - The Riesz Representation Theorem guarantees that in a finite-dimensional Hilbert space (like $ \mathbb{R}^n $ or $ \mathbb{C}^n $), every continuous linear functional can be uniquely represented as an inner product with a fixed vector. That is, for every linear functional $ \phi \in V^* $, there exists a unique vector $ \mathbf{\vec{v}} \in V $ such that:
     \[
     \phi(\mathbf{\vec{x}}) = \mathbf{\vec{v}} \cdot \mathbf{\vec{x}}.
     \]
   - This allows you to treat vectors as linear functionals safely in the context of finite-dimensional vector spaces, using the dot product to convert vectors into functionals.

3. **Wedge Product of Linear Functionals**:
   - In the context of wedge products, when you apply the wedge product to vectors that are assimilated to linear functionals (via the dot product as you described), you essentially recover the wedge product of those functionals. This is because the wedge product of linear functionals (or covectors) naturally corresponds to the alternating multilinear forms you're interested in.

4. **Finite-Dimensional Case**:
   - The reasoning holds specifically because the vector space $ V $ is finite-dimensional, which ensures that every linear functional can be represented by a vector, thanks to the Riesz Representation Theorem. This allows you to safely move between vectors and functionals when dealing with wedge products.

### Conclusion:
Yes, your approach is correct! By assimilating the vector $ \mathbf{\vec{u}}_i $ to the linear functional $ \mathbf{\vec{x}} \mapsto \mathbf{\vec{u}}_i \cdot \mathbf{\vec{x}} $, you're essentially using the Riesz Representation Theorem to move between vectors and functionals. This is valid in finite-dimensional spaces, and it allows you to correctly define the wedge product in terms of determinants as you've described.

This is a very natural approach in finite-dimensional spaces, where the distinction between vectors and linear functionals can be blurred thanks to the inner product structure.

Would you like to explore any further details or examples to solidify this concept? I'm happy to assist!

---

### Further Questions:
1. How would this approach extend or change in infinite-dimensional spaces where the Riesz Representation Theorem still holds but with additional subtleties?
2. Would you like to explore how this assimilation plays out in a concrete example, like in $ \mathbb{R}^3 $ or $ \mathbb{R}^4 $?
3. How does the wedge product behave when applied to linear functionals more abstractly, without relying on inner products?
4. How would you define exterior powers for infinite-dimensional vector spaces?
5. Would you be interested in discussing the dual nature of the exterior algebra and its role in differential forms or physics?

### Tip:
The wedge product of vectors, viewed as linear functionals, plays a crucial role in the context of differential forms in geometry and physics, where it describes things like area and volume elements.

Explore how vectors in finite-dimensional vector spaces can be assimilated to linear functionals using the Riesz Representation Theorem. Learn about the wedge product of linear functionals and its application in mathematical contexts.

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